integration of (ax+by+c)/(ex+fy+g) = dy/dx, where a,c,c,e,f,g are constants

My maths book tells me that if we let X= x+k and y = Y+m, then dY/dX
equals (aX + bY)/(eX +fY) and then proceed as for a homogenous equation. My question is what happened to c and g and how is the new expression homogenous?

ok. Thanks. I still don't understand how removing a constant (c) can be the same as multiplying another constant (a or e) by X over the entire range in which x operates, even if another variable (X) is used. Could you help me, please? What ensures that the new expression tracks the original one?

The substitution required depends on whether the two lines ax+by+c=0 and ex+fy+g=0 are intersecting lines, identical lines, or parallel, non-intersecting lines. If they are identical lines, then the DE simplifies greatly. If they are parallel, non-intersecting lines, then let u = ax+by+c, and rewrite the DE in terms of u and y. The resulting equation will be separable.

Lastly, if the coefficients, when equated to zero, represent intersecting lines, then find the point of intersection (p,q). Define new coordinates u = x-p, v = y-q, and rewrite the DE in terms of these coordinates. The resulting DE will be homogeneous.

visualising your replies

Thanks Ackbeet and Waqarhaider. I tried your suggestions and they worked, but I still cannot visualise or imagine why we let the numerator and the denominator of the original expression equal zero, even though I know the method given uses this trick. Should I just use the method and stop asking questions about the reasons why it works, then? Wouldn't that reduce the intellectual content of the approach somewhat? Suppose I came across another type of problem where a full understanding the method used in the present problem could be of help. Just knowing a method that works without knowing why it works wouldn'd be a lot of use, would it? Can you also please help me understand what's going on with the method you kindly gave me.
As I see it, the original equation describes two 2D surfaces which can be moved up or down according to the value of the constants. These constants are in the way, so to speak, of our calling the expression homogenous, so we use your trick to get rid of them in order to be able to use the standard method for solving such equations. So what are we achieving by making the numerator and the denominator zero? Where's the logic in it?

Thanks Ackbeet and Waqarhaider. I tried your suggestions and they worked, but I still cannot visualise or imagine why we let the numerator and the denominator of the original expression equal zero, even though I know the method given uses this trick. Should I just use the method and stop asking questions about the reasons why it works, then? Wouldn't that reduce the intellectual content of the approach somewhat? Suppose I came across another type of problem where a full understanding the method used in the present problem could be of help. Just knowing a method that works without knowing why it works wouldn'd be a lot of use, would it? Can you also please help me understand what's going on with the method you kindly gave me.
As I see it, the original equation describes two 2D surfaces which can be moved up or down according to the value of the constants. These constants are in the way, so to speak, of our calling the expression homogenous, so we use your trick to get rid of them in order to be able to use the standard method for solving such equations. So what are we achieving by making the numerator and the denominator zero? Where's the logic in it?

Instead of worrying about the general case, attempt a concrete question using the help you've been given so far. Post the question if you can't do it, showing all your work and saying where you get stuck.